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Performance Analysis of Generalized Product Codes with Irregular Degree Distribution

Sisi Miao, Jonathan Mandelbaum, Lukas Rapp, Holger Jäkel, Laurent Schmalen

TL;DR

The paper addresses the decay of intrinsic message-passing decoding for generalized product codes (GPCs) with irregular degree distributions by developing a random hypergraph-based density-evolution framework for the IMP decoding, extending prior results that assumed a regular two-code protection. It models GPCs with fixed interleavers via residual hypergraphs and uses a branching-process argument to derive a tractable decoding-evolution recursion, enabling schedule-aware analysis such as windowed decoding in spatially-coupled settings. The authors apply the DE to BER prediction (with a closed-form BER estimate) and to an Extended Staircase Code (ESC) with irregular degree distributions, showing that increasing the fraction of higher-degree VNs improves decoding thresholds and can lower the error floor, providing a practical design tool for tailoring degree distributions. The framework captures both theoretical performance and practical decoding schedules, offering guidance for constructing efficient high-rate GPCs with manageable complexity and improved mitigation of error floors in hardware-limited regimes.

Abstract

This paper investigates the theoretical analysis of intrinsic message passing decoding for generalized product codes (GPCs) with irregular degree distributions, a generalization of product codes that allows every code bit to be protected by a minimum of two and potentially more component codes. We derive a random hypergraph-based asymptotic performance analysis for GPCs, extending previous work that considered the case where every bit is protected by exactly two component codes. The analysis offers a new tool to guide the code design of GPCs by providing insights into the influence of degree distributions on the performance of GPCs.

Performance Analysis of Generalized Product Codes with Irregular Degree Distribution

TL;DR

The paper addresses the decay of intrinsic message-passing decoding for generalized product codes (GPCs) with irregular degree distributions by developing a random hypergraph-based density-evolution framework for the IMP decoding, extending prior results that assumed a regular two-code protection. It models GPCs with fixed interleavers via residual hypergraphs and uses a branching-process argument to derive a tractable decoding-evolution recursion, enabling schedule-aware analysis such as windowed decoding in spatially-coupled settings. The authors apply the DE to BER prediction (with a closed-form BER estimate) and to an Extended Staircase Code (ESC) with irregular degree distributions, showing that increasing the fraction of higher-degree VNs improves decoding thresholds and can lower the error floor, providing a practical design tool for tailoring degree distributions. The framework captures both theoretical performance and practical decoding schedules, offering guidance for constructing efficient high-rate GPCs with manageable complexity and improved mitigation of error floors in hardware-limited regimes.

Abstract

This paper investigates the theoretical analysis of intrinsic message passing decoding for generalized product codes (GPCs) with irregular degree distributions, a generalization of product codes that allows every code bit to be protected by a minimum of two and potentially more component codes. We derive a random hypergraph-based asymptotic performance analysis for GPCs, extending previous work that considered the case where every bit is protected by exactly two component codes. The analysis offers a new tool to guide the code design of GPCs by providing insights into the influence of degree distributions on the performance of GPCs.
Paper Structure (10 sections, 1 theorem, 6 equations, 7 figures)

This paper contains 10 sections, 1 theorem, 6 equations, 7 figures.

Table of Contents

  1. Introduction
  2. GPCs and Random Hypergraphs
  3. Decoding Evolution of GPCs
  4. Decoding Schedule
  5. Applications of the DE
  6. BER Prediction
  7. Extended Staircase Code with DE
  8. Code Construction
  9. Results
  10. Conclusion

Key Result

Theorem 1

For a GPC, let ${\bm{x}^{(\ell)}=()}$ and $x_i^{(\ell)}$ corresponds to the probability that a bit at position $i\in[I]$ is not recovered after $\ell$ decoding iterations by any of its component codes estimated by its associated IRH assuming ${N^{\mathsf{G}} \rightarrow \infty}$. With $\bm{x}^{(0)}=

Figures (7)

  • Figure 1: Graphical illustration of a 2D PC whose component codes are $(n_\mathsf{c}=2,k_\mathsf{c}=1)$ codes and the process of obtaining the residual graph from its Tanner graph assuming that $\mathsf{v}_1$ and $\mathsf{v}_4$ are erased.
  • Figure 3: An example of one level branching process of one vertex of type $i$. Vertices are depicted as squares with its type in it. Hyperedges are depicted as a circles connecting the vertices. Possible hyperedges are shown in gray. Hyperedges are realized with probability $p$ and depicted in black. Assume that the type $i$ vertex is connected to type $j$ vertices by degree 2 hyperedges and to type $j_1$ and $j_2$ vertices by degree 3 hyperedges.
  • Figure 4: DE and simulation results of the 3D PC with $(127,113,2)$ BCH component codes using $L$ iterations of MF iBDD decoding over a BSC.
  • Figure 5: Graphical illustration of the ESC.
  • Figure 6: Decoding thresholds $p^{\star}$ for ESCs of different rates $R$ based on (a) $(128,120,1)$, (b) $(512,493,2)$, and (c) $(1024,993,3)$ extended BCH component codes using DE.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Example 2
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1